# Simple: What do these ballistic impact numbers represent?

I bought physics for game programmers, found here, today to study ballistic impact, but I'm confused as to where a few numbers are coming from. I'm basically looking to understand what the values represent since the book does a bad job at telling me.

I get that $E_k = \frac{1}{2}mv^2$.

In talking about the F-Formula, developed by Dr. L. Thompson (neither of which I can find anything about) "The equation was presented as a ratio between the plate thickness and the diameter of the projectile."

That equation:

$$\frac{t}{d} = 0.0623 \frac{(mv)^2}{d^3F^2}cos^2(\alpha)$$

I realize that the above was just an example of the equation in the book, but .0623 is pissing me off. I have no idea where they get this from. Probably some number relating to steel armor since that's what he's talking about... so I'll continue...

"The coefficient F is a measure of the penetration resistance of the armor"

That equation:

$$F = 1.8288(\frac{t}{d} - 0.45)(\alpha^2 + 2000) + 12192.$$

Can someone tell me what those numbers are? I don't care where they got them from, but what are they suppose to be? I need to know so I can plug my own in...

Apparently it's also possibly to "rearrange the Thompson formula so that it is in the terms of the minimum projectile energy, $E_k$, necessary to penetrate the armor.

$$Ek = \frac{1}{2}mv^2 = 8.025\frac{(td^2)(F^2)}{cos^2 \alpha}$$

Again... another imaginary number. What is this 8.025??? What should go there?

It says the right hand side of the last equation there tells which factors influence steel armor pen, so I'm guessing the 8.025 is relating to that, but what factor? Thickness?

To further confuse me, it seems like he then tries to take everything he's taught me so far and then simplify it.

Assuming the impact is head on and uses the second to last equation I provided to compute the F coefficient:

$$F = 1.8288(\frac{.01}{.009} - .45)(2000) + 12192 = 14610$$

So then taking that number

$$\frac{1}{2}mv^2 = 8.025 * .01 * .009^2 * 14610^2 = 1387.5 J.$$

It footnotes and says a 9mm bullet has a mass of .0082 kg and a muzzle velocity of 440 m/s. The bullet only does 794 J, so the armor stops the bullet.

I just need someone to tell me what those numbers represent. Much thanks!

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This question is worrisome, since it seems to be about bullets penetrating armor in real life, one must ask--- why do you care? If you are making a game, there is no need to be so realistic, you can just make up armor thickness to stop a given bullet. The numbers you give are useless without units, and the question is unanswerable without the book. – Ron Maimon May 5 '12 at 1:27
By that logic, all physics problems are worrisome. Why do you care that I care? This isn't the Manhattan Project. If you don't want to help people who are asking questions, don't come to stackexchange... Sorry to be a dick, but I thought this was a terribly stupid response. I'll take my question else where, never mind. – Stradigos May 5 '12 at 2:55
Hi Stradigos, and welcome to Physics Stack Exchange! No need to overreact here - all Ron is saying is that it may not be possible to tell what these numbers represent without knowing at least what units are associated with them, and perhaps other information from the book as well. (@Ron, that could have been phrased better...) I'd encourage you to think before deciding not to use this site based on one person's response. – David Z May 5 '12 at 6:01
@Stradigos Are these formulas (For F and t/d) derived from first principles, or is it some empirically obtained formulation from experiments? – Bernhard May 5 '12 at 7:11
They look to be largely empirical. The particular values will obviously depend on the units in use and also a large set of assumptions about the projectile and the "armor", and the nature of the interaction. A search for "Thompson F Formula" turns up some background here: navweaps.com/index_nathan/Hstfrmla.htm – tmac May 5 '12 at 7:19